Preheat oven to 350°F. Butter three 8-inch round cake pans with 2-inch-high sides. Line with parchment paper. Butter paper. Bring 2 cups stout and 2 cups butter to simmer in heavy large saucepan over medium heat. Add cocoa powder and whisk until mixture is smooth. Cool slightly.
Whisk flour, sugar, baking soda, and 1 1/2 teaspoons salt in large bowl to blend. Using electric mixer, beat eggs and sour cream in another large bowl to blend. Add stout-chocolate mixture to egg mixture and beat just to combine. Add flour mixture and beat briefly on slow speed. Using rubber spatula, fold batter until completely combined. Divide batter equally among prepared pans. Bake cakes until tester inserted into center of cakes comes out clean, about 35 minutes. Transfer cakes to rack; cool 10 minutes. Turn cakes out onto rack and cool completely.